## Mathematician's children

A good puzzle using numbers and factors

Here’s a conversation between two mathematicians:

First: “I have three children. The product of their ages is 36. If you sum their ages, it is exactly same as my neighbor’s door number on my left. What are their ages?”
Second: Takes a look at the door number and verifies it. “Well, that data is not enough.”
First: “One more clue. My youngest is the youngest”

Immediately the second mathematician finds out their ages. Can you?

#### Solution

How many possible factors are there for 36 using three numbers?

1. 36 = 3 x 2 x 6
2. 36 = 3 x 3 x 4
3. 36 = 2 x 2 x 9

Of these, only #1 has unique set of ages, i.e., every age is different. This is necessary to validate the second clue of “My youngest is the youngest”.

So, the answer is 2, 3 and 6.

That bit about the neighbor’s door is just a diversion.

1. Alas, not a great puzzle -- because while "1" isn't allowed when coming up with prime factors, it is a perfectly valid age. So there's nothing in the puzzle to prevent the answer from being

1, 2, 18
1, 3, 12
1, 4, 9
1, 6, 6

...unless of course I'm too low on sleep to understand things correctly :)

Posted by: Mike Gunderloy on December 30, 2002 08:56 AM
2. Right on, I overlooked that part! Thanks for the catch.

Posted by: Babu on December 30, 2002 09:28 AM
3. no diversion, there is a bit more logic behind that puzzle.
first you missed one more combination -> 36 = 1 x 1 x 36 is possible as well ;-)

but the trick is the door number. in most cases if you add the ages the result (and door number) is a unique number, with the exception of 13:
13=1+6+6 and 13=2+2+9

so now you should see why the hint was necessary. only in this case the look-up of the door number was insufficient.

the hint shows that there is only "one" smallest age, thus 1,6,6 is the correct solution.

Greetings and a happy New Year from germany :)

Posted by: Gerald on December 30, 2002 03:11 PM
4. Thanks Gerald, you are right.

Posted by: Babu on December 30, 2002 06:34 PM
5. I am an old puzzle fan, I heard about this puzzle 25 years ago, with another story around it. There it was a selling agent who had to solve the puzzle in order to sell his product.

Posted by: Gerald on January 4, 2003 02:19 PM
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